Analysis and Evolution of Balance in Unstable Barotropic Jets

The nature of balance in the atmosphere is of central importance to the dynamics of both the troposphere and the stratosphere, and unbalanced motions such as inertia-gravity waves play a significant role in many aspects of atmospheric behavior. In light of the importance of upper-tropospheric jets for the generation of inertia-gravity waves in the atmosphere, this study examines the evolution of unstable barotropic jets to assess the nature and evolution of balance in these features. This issue is explored using the simplest non-trivial dynamical framework in which balanced and unbalanced flows can coexist, namely the one-layer shallow water equations. In this study, numerical simulations of initially balanced zonal barotropic jets on an f plane are investigated for evidence of the breakdown of balance and the generation of inertia-gravity waves during the life cycles of the instabilities to these jets. In these simulations, the parameters of the basic-state jet (i.e., jet width and speed) are varied systematically in an attempt to elucidate the dependence of balance on the structure and dynamical evolution of the instability. The presence of unbalanced flow, either in numerical simulations or in atmospheric data, is typically inferred via various quantities that provide indirect measures of imbalance, such as the existence of strong ageostrophy, large Rossby and/or Lagrangian Rossby numbers, and large values of horizontal divergence and its material derivative. Along with evaluating these parameters in each simulation, a potential vorticity inversion method is employed to obtain the structure of balanced and unbalanced fields within each simulation. The diagnostic calculations are then compared to the potential vorticity inversion results. Contrasts and comparisons are presented for each of the simulations shown in this study. The simulations consist of an unstable barotropic wave ranging from small (i.e., O(10-1)) Rossby and Froude number to large (i.e., O(1)) Rossby and Froude number. For strong jets, neither the Rossby number nor the Froude number is small compared to unity therefore the applicability of traditional scale analysis is unclear (e.g. Haltiner and Williams, 1980) (i.e., the balance condition is no longer valid and a breakdown of balance should occur). In contrast, the results of the diagnostic calculations and potential vorticity inversions reveal that nonlinear balance is essentially valid for this particular jet profile, even though the Rossby and Froude numbers are O(1) for the strong barotropic jet. Significant inertia-gravity wave structures were not found in any of the cases shown here, which is consistent with the results obtained by several other investigators in their integrations of the shallow-water equations.